As it seems there is little source material to directly answer this question, I have attempted to give as informed an answer as possible, based on material regarding comparable situations.

First of all; your own voice travels not only through air, but also through bone and muscle (bone conduction) resulting in the alteration of vibrations or pitch. In consequence, people (especially kids) are shocked at hearing their own voice, if you had never heard it before, and it was put in a line up, there is a good chance you wouldn't recognise it at first try. This means that in essence, this is no different from how it would have been back in the late 1800's.

However, while there is not much evidence (as far as I can find) on how people reacted on hearing there own voice for the first time in the late 1800's, there is quite a lot of material on public perception regarding other inventions around the same time, such as the telephone.

The 1870's and 1880's being part of the second industrial revolution carried a lot of changes, including electricity, recordings, commercial photography, etc. And it aided in the 'normalisation of technology'. This is not the dark ages, people realised they didn't have to kill every guy with a new toy. When the telephone was invented, the public reaction was comparable to the reaction to most inventions of that time and after: people were intrigued and amazed, some thought it would bring the death of the respectable society (1977). However, through major businesses and a lot of publicity, it slowly seeped into our daily lives. So far, so normal. But, whenever a new technology enters our world, there are stages of anxiety and scepticism (2003) as well. While they are overcome in the long run, individuals will often have to overcome the mental hurdles. When applied to hearing their own voice, this might have resulted in certain people being sceptical of the quality of the recording, thinking the recorded voice was not faithful or the result of a faulty machine. So in short, it wouldn't have been all that different. Some people might have been sceptical - but you would get that reaction still if you did it with a thirteen-year-old for the first time - and in general they would have been just as embarrassed and have just as much fun as us.

Why your voice sounds different: https://www.scientificamerican.com/article/why-does-my-voice-sound-different/

Public reactions to the telephone: Ideology and the Telephone: the Social Reception of a Technology, London 1876-1920 (Jeremy Leon Stein, 1996)

How people deal with change: Innovation, Change Theory and the Acceptance of New Technologies A Literature Review (A. Couros, 2003)